On extremal trees with respect to the F-index
Hosam Abdo1,*, Darko Dimitrov2,3, Ivan Gutman4,5
1Institut für Informatik, Freie Universität Berlin, Germany
2Hochschule für Technik und Wirtschaft Berlin, Germany
3Faculty of Information Studies, Novo Mesto, Slovenia
4Faculty of Science, University of Kragujevac, Serbia
5State University of Novi Pazar, Serbia
*Corresponding author: abdo@mi.fu-berlin.de
Abstract
In a study on the structure–dependency of the total -electron energy from 1972, Trinajstic and one of the present authors have shown that it depends on the sums and , where is the degree of a vertex of the underlying molecular graph . The first sum was later named first Zagreb index and over the years became one of the most investigated graph–based molecular structure descriptors. On the other hand, the second sum, except in very few works on the general first Zagreb index and the zeroth–order general Randić index, has been almost completely neglected. Recently, this second sum was named forgotten index, in short the F-index, and shown to have an exceptional applicative potential. In this paper we examine the trees extremal with respect to the F-index.
Keywords: F-index; general first Zagreb index; molecular trees; topological indices; trees.
1. Introduction
The first and the second Zagreb indices, introduced by Gutman & Trinajstić (1972) are among the oldest graph–based molecular structure descriptors, so-called topological indices (Balaban et al. (1983); Devillers &
Balaban (1990); Gutman (2013)). For a graph with a vertex set and an edge set , these are defined as
and
(1) where the degree of a vertex , denoted by , is the number of the first neighbors of .
Over the years these indices have been thoroughly examined and used to study molecular complexity, chirality, ZE-isomerism and hetero-systems. More about their physico–chemical applications and mathematical properties can be found in Nikolić et al. (2003), Gutman (2013), Furtula et al. (2013), Gutman & Das (2004), Das
& Gutman (2004), Zhou (2004), Gutman et al. (2015), Liu & You (2010) respectively, as well as in the references cited therein. The sum of squares of vertex degrees was also independently studied in quite a few mathematical
papers (Das (2003; 2004); Bell (1992); Caen (1998);
Cioabă (2006); Peled et al. (1999)).
In an early work on the structure–dependency of the total -electron energy by Gutman & Trinajstić (1972) beside the first Zagreb index, it was indicated that another term on which this energy depends is of the form
(2)
For unexplainable reasons, the above sum, except (implicitly) in a few works about the general first Zagreb index (Li & Zhao (2004); Li & Zheng (2005)) and the zeroth–order general Randić index (Hu et al. (2005)) has been completely neglected. Very recently, Furtula and one of the present authors succeeded to demonstrate that
has a very promising applicative potential (Furtula
& Gutman (2015)). They proposed that be named the forgotten topological index, or shortly the F-index.
Before the publication of the paper Furtula & Gutman (2015), the F-index was not examined as such. On the other hand, in some earlier studies on degree–based graph invariants, it appears as a special case.
The general first Zagreb index of a graph is defined as
for (3) and its first occurrence in the literature seems to be the work by Li & Zhao (2004). Observe that . In Li & Zhao (2004) the trees with the first three smallest and largest general first Zagreb index were characterized.
In Li & Zhao (2004) and Li & Zheng (2005) among other things, it was shown that for , the star is the tree on vertices with maximal whereas the path is the tree on vertices with minimal -value. Needless to say that these results directly apply to the F-index More results and information about the general first Zagreb index can be found in Gutman (2014), Hu et al. (2005), Liu & Liu (2010), Li & Zhao (2004), Li &
Zheng (2005), Su et al. (2012), Zhang & Zhang (2006).
The Randić (or connectivity) index was introduced by Randić (1975) and is defined as
(4)
Later, Kier & Hall (1997) have introduced the so-called zeroth–order Randić index
(5)
which was generalized by Li & Zheng (2005). It was named the zeroth–order general Randić index and was defined as
(6)
for any real number . Evidently, . Hu et al. (2005) characterized graphs of maximal degree at most 4 (molecular graphs), with given number of vertices and edges, and with extremal (maximum or minimum) zeroth–order general Randić index. This, again, in the special case renders results for the F-index.
Here, we extend the work on trees with extremal values of F-index, by considering trees with bounded maximal degree. Since the paths are the trees with minimal F-index also in this case, we consider here the characterization of trees with bounded maximal degree that have maximal F-value.
In the sequel we introduce notation that will be used in the rest of the paper. For such that
, we denote by the graph that is obtained by deleting the edge from and by the graph that is obtained by adding the edge to . By we denote the maximal degree of A sequence is graphical if there is a graph whose vertex degrees are , . If in addition , then is a degree sequence.
Furthermore, the notation
means that the degree sequence is comprised of vertices
of degree , where .
A tree is said to be rooted if one of its vertices has been designated as the root. In a rooted tree, the parent of a vertex is the vertex adjacent to it on the path to the root; every vertex except the root has a unique parent. A vertex is a parent of a subtree, if this subtree is attached to the vertex. A child of a vertex is a vertex of which is the parent.
2. Results
First, we characterize trees with maximal degree at most that have maximal F-index.
2.1. Trees with bounded maximal degree
Theorem 2.1. Let be a tree with maximal F-index among the trees with vertices and maximal degree at most . Then the following holds:
(i) If , then contains
vertices of degree and vertices of degree 1.
(ii) Otherwise, contains vertices of degree vertices of degree 1 and one vertex of degree , where is uniquely determined by
and .
Proof. We may assume that is a rooted tree, whose root is a vertex with degree . First, we show that is comprised of vertices of degrees and 1, and in some cases of one additional vertex , with Assume that this is not true and that has more than one such vertex.
Denote by the set of all vertices of whose degrees are different from and 1. Denote by the number of vertices of tree of degree , We assume that for the degrees of the vertices of the order
holds. Let be a child vertex of . Delete the edge and add the edge to , obtaining a tree . After this transformation, the degree of increases by one, while the degree of decreases by one, and the degree set changed into . It holds that
(7) It holds that , with strict inequality if
or . If we choose from a vertex with maximal degree and a vertex with minimal degree, and we repeat the above operation, obtaining a tree with larger F-index. We proceed on iteratively with the same type of transformation until the transformed has cardinality 1 or 0, and thus obtain a contradiction to the assumption that has more than one vertex with degree different from 1 and . It follows that has a degree
sequence or , .
Observe that any transformation on will result in a tree with or will disconnect . Thus, we conclude that must have one of the two above presented degree sequences.
Next, with respect to the cardinality of , we determine the parameters , , and .
Case 1. . In this case the degree sequence of is where the equations
and
(8)
hold. The above equations have the integer solution
(9)
for .
Case 2. . Here the degree sequence of is , with
and
(10)
The above equations give the integer solution
(11)
for .
As straightforward consequence of Theorem 2.1, we obtain the maximal value of the F-index for trees with maximal degree .
Corollary 2.1. Let be a tree with maximal F-index among the trees with vertices and maximal degree at most . Then,
(i) if ,
; (ii) otherwise,
, where is uniquely determined by and
.
Proof. If , by Theorem 2.1,
has degree sequence , where
and . Thus,
(12) Otherwise, has degree sequence , where
. Then,
(13)
2.2. Molecular trees
For the special case of molecular trees, i.e., , we obtain the following result. Recall that such trees provide the graph representation of the so-called saturated hydrocarbons or alkanes (Gutman & Polansky (1986);
Balaban et al. (1983)) and are of major importance in theoretical chemistry.
Theorem 2.2. Let be a molecular tree with maximal F-index among the trees with vertices. Then the following holds:
• If , then contains vertices of degree and vertices of degree 1. Its F-index is
• Otherwise, contains vertices of degree 4, vertices of degree 1 and one vertex of degree , where is uniquely determined by and
. Its F-index is
Proof. For the star maximizes the -index (Li & Zhao (2004)). For and , the possible
degree sequences of are , ,
and . The first of these degree sequences corresponds to the largest F-index, .
Thus, the corresponding degree sequences of for satisfy the theorem.
In the rest of the proof we assume that . First, we show that has maximal degree . A tree with maximal degree we denote by . By Corollary 2.1(i), . Further, we distinguish between two case.
Case 1. .
By Corollary 2.1(i), we have that . Since
, it follows that ,
and by Corollary 2.1(ii), ,
where or . For is satisfied.
Case 2. .
By Corollary 2.1(ii), ,
where or .
Subcase 2.1. .
By Corollary 2.1(i), . For and
it holds that .
Subcase 2.2. .
By Corollary 2.1(ii),
Again, a straightforward calculation yields that .
So, we have shown that for the tree with maximal F-index has maximal degree , and for it has maximal degree . Setting in Theorem 2.1 and Corollary 2.1, we complete the proof.
We would like to note that the results of Theorem 2.2 coincide with the results of Theorem 2.2 in Hu et al.
(2005), pertaining to the zeroth–order general Randić index of a graph, when and .
2.3. Observations and computational results
Degree sequences of trees with bounded maximal degree and maximal are fully characterized by Theorem 2.1. We would like to note that, alternatively, one can obtain a complete characterization by solving the integer linear programming problem presented in Lemma 2.1. We denote the number of vertices of tree of degree by ,
and the number of edges, connecting a vertex of degree with a vertex of degree , by . Lemma 2.1. Let be a tree with maximal F-index among all trees with vertices and maximal degree . Then, the trees with maximal F-index are completely described by solving the maximization problem
For experimental proposes we have solve the above linear programming problem for trees with (molecular trees) and trees with . Needless to say the obtained characterization coincide with that from Theorem 2.1. All trees with maximal F-index of order up to 20 and maximal degree 4 and 5 are given in Figures 1.
and 2., respectively.
Fig. 1. Extremal chemical trees of orders with respect to the F-index.
Fig. 2. Extremal trees of orders with respect to the F-index.
3. Conclusion
The F-index is a graph–based molecular structure descriptor that has an exceptional applicative potential. In this paper we have characterized the extremal trees with vertices and arbitrarily large maximal degree with respect to the F-index. Additionally, we have presented some computational results. The computations were focused on molecular trees and on trees with maximal degree of order up to 20.
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Submitted : 23/09/2015 Revised : 28/11/2015 Accepted : 29/11/2015
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